复二次B样条曲线

本文在复平面单位圆弧上引进了复二次Ｂ样条曲线，讨论了它的一些几何性质．实质上它是分段帕斯卡蜗线段的Ｃ１合成曲线．调整控制点可使某段曲线为圆孤．     

有理二次B—样条曲线的权系数与交比问题

对于控制顶点相同的两条有理二次Ｂ－样条曲线，本文证明了从任一控制顶点引出的射线上给定四点的交比值与射线位置无关的充要条件。对于一条有理二次Ｂ－样条曲线，本文还给出了另一种四个共线点的交比表示，此交比仅与过此顶点的射线位置有关，而与曲线的权系数无关，还给出了此交比在一定意义下为射影不变量的充要条件。     

B—样条曲线的一种高效臬法

为Ｂ－样条曲线及所有导数同时赋值提供一种高效算法，它从最高阶导数赋值开始，把高阶等数于低阶导数的求值。     

Box—样条曲面的正性条件

研究了Ｂｏｘ－样条曲面的控制点与Ｂｏｘ－样条曲面的正性之间的关系，给出了Ｂｏｘ－样条英面正性的必要条件，充分条件，由此我们得到了Ｂｏｘ－样条曲面的单调性条件，推广了Ｗ．ＤＡＨＭＥＮ和Ｃ．Ａ．ＭＩＣＣＨＥＬＬＩ在文「２」中给出的Ｂｏｘ－样条曲面的单高性结论。     

一类有理交错样条曲线的逼近性及形状修改

本文提出一类Ｃ＾２连续的四阶有理交错样条曲线，它能整体或局部逼近于控制多边形。     

一类带参数的有理三次三角Hermite插值样条

给出一种带有参数的有理三次三角Hermite插值样条,具有标准三次Hermite插值样条相似的性质.利用参数的不同取值不但可以调控插值曲线的形状,而且比标准三次Hermite插值样条更好地逼近被插曲线.此外,选择合适的控制点,该种插值样条可以精确表示星形线和四叶玫瑰线等超越曲线.     

Box－样条曲面的正性条件

研究了Ｂｏｘ－样条曲面的控制点与Ｂｏｘ－样条曲面的正性之间的关系．给出了Ｂｏｘ－样条曲面正性的必要条件、充分条件．由此我们得到了Ｂｏｘ－样条曲面的单调性条件，推广了Ｗ．ＤＡＨＭＥＮ和Ｃ．Ａ．ＭＩＣＣＨＥＬＬＩ在文［２］中给出的Ｂｏｘ－样条曲面的单调性结论     

与给定多边形相切的C2和C3 广义 Ball闭曲线

B样条曲线和Bezier曲线在CAD、CAGD及逼近中非常有用,例如,在服装CAD纸样的设计过程中,通常先根据尺寸要求确定控制点,再根据控制点用直线段绘制样片的大体轮廓,即绘制多边形,在多边形内用直线和曲线绘制封闭的图形,由于样片的形状不规则,即构成样片的曲线比较复杂,为了使曲线光滑、丰满、有弹性,经常遇到曲线与控制多边形相切     

STUDY OF THE SMOOTH DEGREE OF THE CLOSED C-B（E）ZIER AND B-TYPE SPLINE CURVE

本文研究了与多边形相切的样条曲线的构造方法和基本属性问题,给出了曲线光顺度的一般定义和计算方法.利用该方法对分段C-Bézier曲线、4-5-5-4次交错B-样条曲线和3阶B样条曲线的光顺度进行计算,获得了3阶B样条曲线最为光顺的结果.     

一种四次有理插值样条及其逼近性质

1引言有理样条函数是多项式样条函数的一种自然推广,但由于有理样条空间的复杂性,所以有关它的研究成果不象多项式样条那样完美,许多问题还值得进一步的研究．近几十年来,有理插值样条,特别是有理三次有理插值样条,由于它们在曲线曲面设计中的应用,已有许多学者进行了深入研究,取得了一系列的成果(见[1]-[7])．但四次有理插值样条由于其构造所花费的计算量太大以及在使用上很不方便而让人们忽视了其重要的应用价值,因此很少有人研究他们．实际上,在某些情况下四次有理插值样条有其独特的应用效果,如文[8]建立的一种具有局部插值性质的分母为二次的四次有理样条,即一个剖分     

Bézier曲线降多阶逼近的一种方法

文献[1,2]讨论了Bezier曲线一次降多阶逼近问题,得到了很好的结果.文献[1]利用广义逆矩阵得到不保端点插值的降多阶逼近曲线的控制顶点的表达式.但却没有得到带端点任意阶插值条件的降多阶逼近曲线的控制顶点的表达式.文献[2]得到了带端点任意阶插值的降多阶逼近曲线的控制顶点的解析表达式.本文首先给出两Bezier曲线间距离的定义;然后根据降阶曲线与原曲线间的距离最小,分别得到了用矩阵表示的不保端点插值和保端点任意阶插值的降多阶逼近曲线的控制顶点的显示表达式.所给数值例子显示,用本文方法得到的降多阶逼近曲线对原曲线有很好的逼近效果.     

Cn-连续的B-型样条曲线及其局部性、插值与逼近

§ 1　IntroductionB-spline curve plays an important role in CAGD.Cn-continuous B-spline curve of de-gree n+ 1 has local support of length n+ 2 [1 ] .In order to modify the shape of a curve insmaller range,the length of its supportshould be as shortas possible.The interpolation ofa set of given data points with the B-spline curve generally needs to solve a system of lin-ear equations or to insert some additional control points[2— 5] .Besides,the global approxi-mation of the control polygon …     

一类参数曲线正则性的判别

Bézier曲线的正则性,完全由它的控制顶点决定.理想的情况是由Bézier曲线的控制顶点的几何关系,就可以判断它的正则性.本文由Bézier曲线的导矢曲线在[0,1]不等于零这些代数条件,推导出了与之等价的Bézier曲线的控制顶点之间的几何关系,即只需知道顶点之间的相对位置或计算相邻线段的斜率就可快速判断Bézier曲线的正则性.最后给出了数值例子.     

空间有理三次Bezier曲线的射影变换和权系数的几何性质

本文讨论了空间有理三次Ｂｅｚｉｅｒ曲线的射影变换和权系数的一系列几何性质。其权系数组成构成了控制四顶点基下的权心的齐次坐标;权心是六个特殊平面的公共交点。含权心和曲线“肩点”的某四个共线点之比恒为常数３;权心可作为有理曲线所在射影坐标系的单位点;此有理曲线是对应整有理曲线在射影变换下的象,此变换把控制四面体的形心映为权心;权系数是此射影变换的特征值（差－常数因子）;权系数是变换前后两曲线上对应点关     

Offset approximation based on reparameterizing the path of a moving point along the base circle

This paper presents a novel algorithm for planar curve offsetting. The basic idea is to regard the locus relative to initial base circle, which is formed by moving the unit normal vectors of the base curve, as a unit circular arc first, then accurately to represent it as a rational curve, and finally to reparameterize it in a particular way to approximate the offset. Examples illustrated that the algorithm yields fewer curve segments and control points as well as C^1 continuity, and so has much significance in terms of saving computing time, reducing the data storage and smoothing curves entirely.     

与给定多边形相切的可调广义Ball闭曲线

讨论了与给定控制多边形相切的分段三次、五次和六次可调广义Ball曲线的构造方法,所构造的曲线分别是C1,C2和C3连续的,而且对切线多边形是保形的.曲线上的所有广义Ball曲线段的控制点由切线多边形的顶点直接计算产生.给出了在保持公共连接点处相应连续的情况下,内控制点的活动范围.曲线可以在一定范围内做局部修改.计算实例...     

Modified Schwarz–Christoffel mappings using approximate curve factors

The Schwarz–Christoffel mapping from the upper half-plane to a polygonal region in the complex plane is an integral of a product with several factors, where each factor corresponds to a certain vertex in the polygon. Different modifications of the Schwarz–Christoffel mapping in which factors are replaced with the so-called curve factors to achieve polygons with rounded corners are known since long times. Among other requisites, the arguments of a curve factor and its correspondent scl factor must be equal outside some closed interval on the real axis.In this paper, the term approximate curve factor is defined such that many of the already known curve factors are included as special cases. Additionally, by alleviating the requisite on the argument from exact to asymptotic equality, new types of curve factors are introduced. While traditional curve factors have a C¹ regularity, C^∞ regular approximate curve factors can be constructed, resulting in smooth boundary curves when used in conformal mappings.Applications include modelling of wave scattering in waveguides. When using approximate curve factors in modified Schwarz–Christoffel mappings, numerical conformal mappings can be constructed that preserve two important properties in the waveguides. First, the direction of the boundary curve can be well controlled, especially towards infinity, where the application requires two straight parallel walls. Second, a smooth (C^∞) boundary curve can be achieved.     

B—样条逼近曲线的应用

引进了一种构造曲线的逼近技术，它放松了曲线应包含所有数据点这一严格要求。为了度量一曲线能逼近已给数据多边形的好坏，使用了移动控制点的概念。     

非线性回归方法的应用与比较

比较了非线性回归3种方法的数学原理:曲线直线化方法、非线性最小二乘方法、近似非线性法.说明了用方差分析确定回归模型的统计学意义、用决定系数R2描述曲线的拟合效果的理论依据.通过对同一问题用3种方法分析得出结论:非线性回归与近似非线性拟合方法决定系数相近(0.9966与0.9965),而曲线直线化决定系数为0.9738.因为近似非线性拟合方法无需选初值.建议应用近似非线性拟合方法.     

Circular spline fitting using an evolution process

We propose a new method to approximate a given set of ordered data points by a spatial circular spline curve. At first an initial circular spline curve is generated by biarc interpolation. Then an evolution process based on a least-squares approximation is applied to the curve. During the evolution process, the circular spline curve converges dynamically to a stable shape. Our method does not need any tangent information. During the evolution process, the number of arcs is automatically adapted to the data such that the final curve contains as few arc arcs as possible. We prove that the evolution process is equivalent to a Gauss-Newton-type method.